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Final Exam PART A, Dec., 2005 \hfil SHOW ALL WORK \hfil
\vskip -10pt
Math 25 Calculus I \hfil Either circle your answers or place on answer line.



All problems required on this part of the exam.



1.) Suppose $f(x) = \sqrt{4x -3}$ and $g(x) = x$.

$\sqrt{4x -3} = x$

$(4x -3) = x^2$

$x^2 - 4x + 3 = 0$

$(x - 1)(x - 3) = 0$.  Hence $x = 1, 3$

Take $2 \in (1, 3)$

$x = 2:  \sqrt{4x -3} = \sqrt{5}$

$x = 2:  x = 2$


$\sqrt{5} > 2$.  Hence $\sqrt{4x -3} > x$ on (1, 3)

[3]~ 1a.)  Set up, {\bf but do NOT evaluate}, an integral for the area of the 
region enclosed by $f$ and $g$.

height = $\sqrt{4x -3} - x$, width = $dx$

Area = $\int_{1}^3 (\sqrt{4x -3} - x)dx$

[4]~ 1b.) Set up, {\bf but do NOT evaluate}, an integral for the volumne of the solid obtained by rotating the region bounded by the curves $f$ and $g$ about the line $x = 8$ (hint:  use cylindrical shells).

Area = $2\pi rh$, $h = \sqrt{4x -3} - x$, $r = 8-x$,

Area = $2\pi(8-x)(\sqrt{4x -3} - x)$

width or thickness of cylindrical shell = $dx$.

Volume = $2\pi \int_{1}^3 (8-x)(\sqrt{4x -3} - x)dx$



[4]~ 1c.) Set up, {\bf but do NOT evaluate}, an integral for the volumne of the solid obtained by rotating the region bounded by the curves $f$ and $g$ about the line $y = -4$ (hint:  use washers).


Area = $\pi(R^2 - r^2)$, $R = \sqrt{4x -3} - (-4)= \sqrt{4x -3} + 4$, $r = x  - (-4) = x + 4$

width or thickness of washer = $dx$.


Volume = $\pi \int_{1}^3 [(\sqrt{4x -3} + 4)^2 - (x + 4)^2]dx$


\eject


[1]~ 2.)  If $h(x) = 3x^2$, then the slope of the tangent line at the point (1, 3) is $\underline{6}$

$h'(x) = 6x$, $h'(1) = 6(1) = 6$

[10]~ 3.)  Find the derivative of $g(x) = ln( {4e^{x^3} - 2 \over 5x +2})$

$ln( {4e^{x^3} - 2 \over 5x +2}) = ln( {4e^{x^3} - 2(5x +2)^{-1} })$

\centerline{Answer 3.) $\underline{
{1 \over {4e^{x^3} - 2 \over 5x +2}}[4e^{x^3}(3x^2) + 2(5x +2)^{-2}(5)]
}$}

\eject

4.)  Find the following integrals:

[10]~ a.) $\int_1^e {ln(x^2) \over x} dx$ $= \underline{\hskip 2in}$
[SIMPLIFY your answer]

\vfill

[3]~ b.)  $\int {3 \over \sqrt{1 - x^2}} dx$ $= \underline{\hskip 2in}$


\vskip 40pt

[10]~ 5.)  Find the following limit  (SHOW ALL STEPS):  
$lim_{x \rightarrow \infty} x^4 e^{-3x^2}$ $= \underline{\hskip .8in}$


\vfill

\eject
\vfill
\s\s

6.) Find the following for $f(x) =  x^{3 \over 2} - 2x^{1 \over 2}
= x^{1 \over 2} (x - 2)$
 (if
they
exist; if they don't exist, state so).  Use this information to graph $f$.





Note $f'(x) = {3 \over 2}x^{1 \over 2} - x^{-{1 \over 2}}
= x^{-{1 \over 2}} ({3 \over 2}x - 1)$
 and $f''(x) = {3 \over 4}x^{-{1 \over 2}} - {-1 \over 2}x^{-{3 \over 2}}
= x^{-{3 \over 2}} ({3 \over 4}x + {1 \over 2})$

%%Is $f$ even, odd, periodic?  What is the domain and range of $f$?
\vskip -2pt

[1] 6a.) critical numbers: $\underline{\hskip 1.5in}$


[1] 6b.) local maximum(s) occur at $x = \underline{\hskip 1.5in}$

[1] 6c.) local minimum(s) occur at $x = \underline{\hskip 1.5in}$

[1] 6d.)  The global maximum of $f$ on the interval [0, 5] is
$\underline{\hskip .5in}$ and
occurs at $x = \underline{\hskip 1in}$

[1] 6e.)  The global minimum of $f$ on the interval [0, 5] is
$\underline{\hskip .5in}$ and
occurs at $x = \underline{\hskip 1in}$



[1] 6f.) Inflection point(s) occur at $x = \underline{\hskip 1.5in}$


[1] 6g.)  $f$ increasing on the intervals $\underline{\hskip 1.5in}$

[1] 6h.)  $f$ decreasing on the intervals $\underline{\hskip 1.5in}$

[1] 6i.)  $f$ is concave up on the intervals $\underline{\hskip 1.5in}$

[1] 6j.)  $f$ is concave down on the intervals$\underline{\hskip 1.5in}$


[1] 6k.) What is the domain of $f$? $\underline{\hskip 1.5in}$

[1] 6l.) What is the range of $f$? $\underline{\hskip 1.5in}$


\vskip -4pt
[4] 6m.)  Graph $f$
\vskip -8pt
\includegraphics[width=48ex]{grapht}

\eject


Final Exam PART B, Dec., 2005 \hfil SHOW ALL WORK \hfil
\vskip -10pt  
Math 25 Calculus I \hfil Either circle your answers or place on answer line.


Choose 4 out of the following 5 problems:
{\bf Clearly indicate which 4 problems you choose.}  Each problem is worth 10 points
You may do all the problems for up to five points extra credit.

I have chosen the following 4 problems: $\underline{\hskip 3in}$

A.)  Find the derivative of $f(x) = (x+1)^{x^3}$

\vfill
\centerline{Answer A.) $\underline{\hskip 4in}$}


\s\s\s


B.)  Express the following integral as a limit of Riemann sums.  Do not evaluate the limit:  $\int_0^5 x^2ln(4x + 1) dx$.

\vskip 70pt
\centerline{Answer B.) $\underline{\hskip 4.5in}$}
\eject

C.)  Find the horizontal asymptotes of $f(x) = {\sqrt{x^2 - 3} \over 2x - 1}$

\vfill
\centerline{Answer C.) $\underline{\hskip 4.5in}$}
\eject

D.) A plane flying horizontally at an altitude 400km and at a speed of 2000 km/hr.  passes directly over a radar station.  Find the rate at which the distance from the plane to the station is increasing when it is 500km from the station.

\vfill
\centerline{Answer D.) $\underline{\hskip 4in}$}
\eject

E.)  A cylindrical can without a top is made to contain 100 cm$^3$ of 
water.  Find the dimensions that will minimize the cost of the metal to make the can.  Explain why your answer is optimal.


\vfill
\centerline{Answer E.) $\underline{\hskip 4in}$}

\eject

Final Exam Part C, Dec., 2005 \hfil SHOW ALL WORK \hfil
\vskip -10pt
Math 25 Calculus I \hfil Either circle your answers or place on answer line.


Note the proof problems on this page are completely optional.  You may choose to
prove one (and only one) of the following statements.  If you choose to do one of
the
following problems, it can replace your lowest point problem in the optional
section
OR 80\% of your lowest point problem in the required section.  If you chose to
do one
of the following problems, clearly indicate your choice.

I have chosen the following problem: $\underline{\hskip 3in}$



I.)  Prove 
$f$ differentiable implies $f$ continuous.



II.)  State and prove the Extreme Value theorem (you may use the phrase ``other case is similar'' in your proof).



\end